Ellipse Formula

Ellipse is the set of all points in a plane where the sum of the distances to two fixed points (foci) is constant.

The Formula

(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
Foci: c2=a2−b2c^2 = a^2 - b^2 (where a>ba > b). Eccentricity: e=cae = \frac{c}{a} (with 0≤e<10 \leq e < 1).

When to use: Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

Quick Example

x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1 has center (0,0)(0,0), semi-major axis a=5a = 5 (horizontal), semi-minor axis b=3b = 3 (vertical). Foci at (±4,0)(\pm 4, 0) since c=25−9=4c = \sqrt{25 - 9} = 4.

Notation

aa = semi-major axis (longer), bb = semi-minor axis (shorter), cc = distance from center to each focus.

What This Formula Means

The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.

Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

Formal View

{(x,y)∣d((x,y),F1)+d((x,y),F2)=2a}\{(x,y) \mid d((x,y), F_1) + d((x,y), F_2) = 2a\}; standard form (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 with c2=a2−b2c^2 = a^2 - b^2, eccentricity e=ca<1e = \frac{c}{a} < 1

Worked Examples

Example 1

easy
Find the lengths of the semi-major and semi-minor axes of the ellipse x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1.

Answer

a=5 (semi-major),b=3 (semi-minor)a = 5 \text{ (semi-major)}, \quad b = 3 \text{ (semi-minor)}

First step

1
The standard form is x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 where a>b>0a > b > 0.

Full solution

  1. 2
    Here a2=25a^2 = 25 and b2=9b^2 = 9, so a=5a = 5 and b=3b = 3.
  2. 3
    The semi-major axis has length a=5a = 5 (along the xx-axis) and the semi-minor axis has length b=3b = 3 (along the yy-axis).
An ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 has semi-major axis aa (the larger denominator's square root) and semi-minor axis bb. The major axis lies along whichever variable has the larger denominator.

Example 2

medium
Find the foci of the ellipse x216+y225=1\frac{x^2}{16} + \frac{y^2}{25} = 1.

Example 3

medium
Why is c2=a2−b2c^2=a^2-b^2 (not a2+b2a^2+b^2) for an ellipse?

Common Mistakes

  • Using a2+b2a^2+b^2 for the foci - an ellipse uses c2=a2−b2c^2=a^2-b^2; the plus version is for hyperbolas.
  • Assuming aa is always under xx - the major axis lies under the LARGER denominator, which may be yy.
  • Confusing it with a circle - unequal denominators mean an ellipse, not a circle.

Why This Formula Matters

Planetary orbits, whisper galleries, and lithotripsy all rely on the constant-sum-of-distances property; reading aa, bb, and the foci from standard form is the core conic skill that separates an ellipse from a circle or hyperbola. The focus relation c2=a2−b2c^2=a^2-b^2 (a MINUS) is the detail students most often swap with the hyperbola's plus. Recognizing it by "Are both squared terms positive, added, with different denominators equaling 1?" — rather than by familiar numbers — is what lets a student tell it apart from circle and hyperbola and focus relation mix-up in a mixed problem set.

Frequently Asked Questions

What is the Ellipse formula?

The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.

How do you use the Ellipse formula?

Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

What do the symbols mean in the Ellipse formula?

aa = semi-major axis (longer), bb = semi-minor axis (shorter), cc = distance from center to each focus.

Why is the Ellipse formula important in Math?

Planetary orbits, whisper galleries, and lithotripsy all rely on the constant-sum-of-distances property; reading aa, bb, and the foci from standard form is the core conic skill that separates an ellipse from a circle or hyperbola. The focus relation c2=a2−b2c^2=a^2-b^2 (a MINUS) is the detail students most often swap with the hyperbola's plus. Recognizing it by "Are both squared terms positive, added, with different denominators equaling 1?" — rather than by familiar numbers — is what lets a student tell it apart from circle and hyperbola and focus relation mix-up in a mixed problem set.

What do students get wrong about Ellipse?

The procedure for ellipse is the easy part; the trap is using a2+b2a^2+b^2 for the foci. Asking "Are both squared terms positive, added, with different denominators equaling 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Ellipse formula?

Before studying the Ellipse formula, you should understand: equation of circle.

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